Digital reconstruction of bright field phase contrast images from high resolution electron micrographs

Ultramicroscopy 5 (1980) 479-503 © North-Holland Publishing Company

DIGITAL RECONSTRUCTION OF BRIGHT FIELD PHASE CONTRAST IMAGES FROM HIGH RESOLUTION ELECTRON MICROGRAPHS * E.J. K I R K L A N D and B.M. SIEGEL School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853. USA and N. UYEDA and Y. FUJIYOSHI Institute for Chemical Research, Kyoto University, U/i, Kyoto-Fu 611, Japan Received 15 April 1980

The theory of bright-field image formation of thin specimens in a conventional transmission electron microscope is presented. The recorded image contrast is shown to be predominantly linear in the electron atom scattering amplitudes which are in general complex (possessing phase and amplitude). A linearized image model describing multiple images of varying defocus (defocus series) is derived. Image degradation is characterized by an instrumental transfer function (including spherical aberration, defocus and partial coherence), a finite signal-to-noise ratio and a Debye-Waller temperature factor. Using the minimum mean square error criterion, a new method of image reconstruction to recover the real and imaginary parts of the ideal phase contrast image from a defocus series is derived. This new method of image reconstruction reduces to the well known Wiener filter in the appropriate limiting conditions. A defocus series of micrographs taken on the Kyoto 500 keV electron microscope using a radiation damage resistent specimen of chlorinated copper phthalocyanine is processed. The signal-to-noise ratio of this series is found to be approximately 10. A resolution of ~2 A is apparent in the unprocessed images. The complex image reconstructed from this defocus series shows increased resolution in the real part of the image (~1.4 A) and increased heavy/light atom contrast in the imaginary part of the image.

1. Introduction

higher acceleration voltage) or by computer correction o f the aberration function, all o f which impose large practical problems. Scherzer [2,3] has also discussed the limits o f resolution imposed by radiation damage. The radiation dose tolerable by most specimens (especially organic specimens) will not produce an image with sufficient signal to noise (S/N) (due to statistical electron counting noise) and contrast to reproduce high resolution information. Nathan [4], Unwin and Henderson [5], and Kuo and Glaeser [6] have used signal averaging over many identical areas of a periodic specimen to increase the average SIN while maintaining a low radiation dose per unit volume o f the specimen, to lessen the effects o f radiation damage. Ottensmeyer [7] and more recently Frank et al. [8] have extended this signal averaging technique to nonperiodic specimens. In the work described below a combination o f high voltage (500 kV), signal aver-

Past attempts to image atomic structure using bright field phase contrast electron microscopy (CTEM) have been thwarted by the large aberrations present in the electron lenses used to form images and the high sensitivity o f most specimens to radiation damage. Scherzer [1 ] has shown that the principal aberration affecting the resolution o f the electron microscope in the bright field mode is spherical aberration (Cs). The minimum resolvable separation distance do between two objects is proportional to (Csk3) 1/4, where k is the wavelength o f the imaging electrons. This limit may be extended by decreasing C.~ or ~ (by using a * A short summary of this paper was presented at the 37th Annual Meeting of the Electron Microscopy Society of America (EMSA 1979) by E.J. Kirkland. 479

480

E.J. Kh'kland et aL / Dt~eitalreconstnlction o f phase contrast hnages

aging, and a new method of digital image reconstruction from a defocus series has been used to overcome enough of the image degradation caused by the above effects to visualize atomic structure in an organic specimen of chlorinated copper phthalocyanine (40 A. thick crystal). Although the elusive goal of single atom detection has not yet been attemped, atomic resolution has been realized in the reconstructed images. Zeitler and Olsen [9], have pointed out that the ideal unaberrated phase contrast image in high resolution bright field electron microscopy is a complex quantity. Bright field electron micrographs of thin specimens are approximately linear in the electronatom elastic scattering amplitudes. The optical theorem of quantum mechanics derived by Glauber and Schomaker [10] requires that these scattering amplitudes be complex quantities to conserve the total number of particles. The first Born approximation, however, yields a real value for the elastic scattering amplitude and neglects the phase change of the scattered electron wavefunctions. A more rigorous calculation of the scattering anaplitudes [9,10] shows that the phase change is a strong function of atomic number and is negligible for carbon but large for uranium. More importantly for thicker specimens the relative displacement of the atoms parallel to the optical axis of the microscope causes the scattered wave to be complex due to the different defocus values of each of the atoms. Therefore the ideal unaberrated image of an arbitrary thin specimen must be considered as a complex quantity. Frank [11,12], Hoppe [13] and Reimer and Gilde [14] have also discussed image formation using complex scattering amplitudes. The total phase contrast transfer function (CTF) of the electron microscope is also a complex quantity. Depending upon the CTF of the microscope, different parts of the Fourier spectrum of the real and imaginary parts of the ideal complex image are exchanged (and/or modulated) and the real part of the resulting image is recorded. The recorded real inaage has part of the spectrum of the real part of the ideal image and part of the spectrum of the imaginary part of the ideal image. If several micrographs are recorded at several appropriately different defocus values (different CTFs), these micrographs (defocus series) together contain enough information, in one form or another, to reconstruct the whole ideal unaberrated complex image. In essence, the transfer

function of the microscope can be used to channel parts of the spectrum of the imaginary part of the ideal complex image into the recorded real part of the aberrated complex image so that the imaginary part of the image may also be recorded (although in an unrecognizable form). However, a computer is required to sort out what belongs in the real part and what belongs in the imaginary part of the ideal complex image. The CTF also has the annoying property of transferring some spatial frequencies with positive contrast and others with negative contrast. The computer can also correct for this contrast reversal. A method of digital image reconstruction (or restoration) to recover the full complex image from a defocus series of bright field electron micrographs of thin specimens was first proposed a decade ago by Schiske [ 15]. This method was studied extensively and applied to actual electron micrographs of stained DNA by Frank [11112] a few years thereafter. Schiske's original method [15] is simply the ordinary least squares method (OLS) commonly used in curve fitting of experimental data [ 16] applied successively to each point of tile Fourier transforms of the original micrographs. As will be seen below, this method is appropriate only in regions where the transfer functions and signal-to-noise ratios of the input micrographs are large. The use of additional statistical information about the micrographs (such as S / N ) can greatly improve the results of image reconstruction. Electron micrographs normally obtained in practice have a low S I N (typically 1-10). Because the noise is a significant part of the micrograph and is random, it is necessary to consider the statistical properties of the noise. A method of image reconstruction must be found that is optimized in a statistical sense. A "criterion of optimality" that uses the statistical properties of the noise and is in common use called the minimum mean square error (MMSE) criterion [17] has been adopted here. The well known Wiener inverse f'dter [17] is an example of a method of image restoration from~ single input image considering the statistical properties of the noise using the MMSE criterion. Schiske has also proposed a method of digital image reconstruction from a defocus series using the MMSE criterion [18]. This method however requires the cross correlation between the real and imaginary parts of

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E.J. Kirkland et al. /Digital reconstnwtion o f phase contrast images

the ideal unaberrated image and cannot be'used to reconstruct an unknown complex image. The sensitivity to small errors in the electron optical parameters determining the image degradations and the sensitivity to misalignment of the input micrographs as well as the optimal selection of input micrographs for a simplified version of this method have been discussed analytically by Kirkland and Siegel [19]. In this paper Schiske's original idea [15] is generalized to conform to the MMSE criterion. Using this new method an unknown complex image may be reconstructed from a defocus series of low SIN micrographs (SIN ~ 10). It is also interesting to note (as will be shown below) that this new method reduces to the well known Wiener inverse filter, Schiske's original method [15], and the simplified version of Schiske's MMSE solution [ 18] under the appropriate limiting conditions. As discussed in detail by Frank [11,12] it is highly desirable to recover the ideal unaberrated complex image. The imaginary part of this image should show a high contrast between heavy and light atoms. This high contrast should permit single heavy atom detection in stained biological specimens supported on thin carbon films. Normal "best focus" bright field electron micrographs have a contrast that varies only slightly with atomic number causing single heavy atoms to be mostly hidden in the background image of the carbon supporting fdm. Recovery of the imaginary part of the ideal unaberrated image should permit partial imaging of biological structures stained with heavy marker atoms (such as the uranyl-stained DNA specimens used by Frank [ 11 ]) by increasing the contrast of these heavy atoms and allowing visualization of their positions. However, radiation damage will still be a major limiting factor [2,3]. Also modeling the ideal image as a complex quantity is a more appropriate model and results in reconstructed images of higher quality. Recently several books and review articles relevant to this paper have appeared. References [17,20,21] treat general methods of digital image processing while references [22,23] specifically treat digital image processing of electron micrographs. 2. Image model Before image processing can be discussed it is

necessary to obtain an analytical description of the image. Many authors [1,24-26] have discussed analytical image models. In what follows, only the continuous form of image analysis and processing will be presented. However, in actuality the image is represented in the computer as a finite number of discrete elements called pixels. A typical image is 128 × 128 = 16,374 complex numbers or pixels (one complex number is four 16-bit words). See also refs. [20-23]. A thin specimen is placed in the electron microscope in a plane perpendicular to the optical axis of the microscope. The specimen is illuminated by a coherent or partially coherent beam of electrons with wavefunction I~in c. These electrons are traveling along the optical axis and are uniform in intensity across the area of the specimen. Upon passing through the specimen and imaging lenses of the microscope the electron wavefunction, t~imag(X ) at the image plane is ~imag(X) = ~ u ( X ) + I~s(X ) .

(1)

~bu(X) and ~s(X) represent the unscattered and scattered waves respectively, x is a two-dimensional position vector in the image plane perpendicular to the optical axis of the electron microscope. The recorded image intensity distribution g(x) is if(X) = [~imag(X)[ 2 + n(x) * + *+

=l~ul2+¢uffs

~bu~s

Iffsl2+n(/)

(2)

A superscript asterisk denotes complex conjugation.

n(x) represents the noise content of the micrograph and is regarded as a random variable with a mean value of zero. To ensure that the image is in the linear, kinematical (i.e., no multiple scattering) region, the following conditions are required of the specimen: I~inc [2 I¢ql 2 [~u [2

1 - exp(--POTt) < < ~

1-exp(-POet) ( S ) -l exp(--POTt) < < '

,

(3a)

(3b)

where t = thickness of specimen, SIN = average signalto-noise ratio of image, p = atomic density of specimen, (re = elastic electron-atom scattering cross section, ai = inelastic electron-atom scattering cross section,

E.J. Kirkland et al. /Digital reconstruction o f phase contrast bnages

482

a n d a T = o i + (7e = total electron-atom scattering cross section. If more than one atomic species is present in the specimen, then POT and POe are replaced by sums over the different species present. The quantities represented on the left sides of conditions (3)represent small signals that are dominated by the noise content of the image and can be ignored with respect to the noise term, n(x) (or conversely these signals may in some sense be thought of as contributing to the noise). Condition (3a) insures that I~ku12 is roughly constant over the image area. The first term on the right-hand side ofeq. (2), Ifful 2, can then be ignored because it contains no structure information. Furthermore, because ~ku (and flu) are approximately constant over the image area, the interference terms ~u~s and ~0u~ks are approximately linear in the desired structure information ~s and ffs • Condition (3a) also ensures that the scattering is kinematical and not dynamical (i.e., no multiple scattering), Condition (3b) allows the so-called quadratic term I ~s 12 in (2) to be ignored. This greatly simplifies the following analysis by making the imaging process approximately linear. Together these conditions can only be satisfied by weakly scattering thin specimens. When the linear-kinematic conditions (eq. (3)) are satisfied the recorded image intensity distribution is

g(x) ~ ~bu(X) ~bs(X) + ~bu(X)

~s

(x) + n(x)

(4)

(to within an additive constant), and the unscattered (~0u) and scattered (ffs) waves are approximately d/u ~

Vine/M,

X exp(27rik, x/M) dk.

(5a)

(5b)

M is the electron optical magnification of the image and ~, is the wavelength of the incident electrons. The relationship between the vectors is shown in fig. 1. kinc and kscatt are the wavevectors of the incident and scattered waves respectively and q is their difference. x is a two-dimensional vector in the image plane and k is the corresponding two-dimensional frequency vector of the Fourier transform of the image. Sin t~ ~ ~lkJ = X/A = a is the electron-atom scattering angle

~ s c o

I tt~ ~"

in z direction) \

EWALD ///SPHERE

~ 7 "~"~ ~

/~

/

.~ t

~ ~~'_ ~

2

L'tC 2x

,|_

(image spatial frequency vector)

PLANE PARALLEL TO OBJECT PLANE (kx-ky PLANE)

Fig. 1. Vector diagram showing the relation between the incident electron wavevector kinc, the scattered electron wavevector kscat t and the spatial frequency vector k of the recorded image. The approximate magnitudes of some vectors are shown to lowest order in the scattering angle a.

where A = 1/Ik] is the spatial wavelength of the Fourier components of the image. H(k) is the transfer function of the microscope. The expression in brackets ( ... ) is the scattering amplitude of the specimen. ru is the three-dimensional position vector of one atom in the specimen and fu(a) is the corresponding complex scattering amplitude of that atom for elastic electron-atom scattering. The summation runs over all atoms in the specimen. Because the Ewald sphere is curved q has a non-zero z component. This causes the z component of the atomic position r u to enter as an additional defocus factor. This may make the interpretation of the image of a specimen more than a few atoms thick very difficult. Inelastic scattering has been ignored in the image model (eq. (5)) although its relative contribution can be large for specimens with low atomic number. Because of the large chromatic aberration of electron lenses the inelastically scattered electrons form a diffuse image of low resolution. Frank [12] has pointed out that the inelastically scattered electrons contribute to the image irrespective of chromatic aberration by removing electrons from the unscattered wave ~u- However, the inelastic scattering is predominately at very low angles (for instance see Misell and Burge [27]) and thus will mostly affect the low resolution information. This inelastic image component may be minimized by simple high pass filtering and will be

E.J. Kirkland et.al. /Digital reconstruction of phase contrast images assumed to be part of the image noise n(x) although it is not strictly uncorrelated with the elastic image (eq. (5)). Scherzer [ 1] derived the instrumental transfer function of the microscope for coherent illumination. Frank [28] and Wade and Frank [29] have recently derived an analytical expression for the instrumental transfer function including the effects of partial coherence and chromatic aberration defocus spread, which has been adopted here. Also, at room temperature the atoms of the specimen vibrate about their normal positions and attenuate the very high resolution information. To allow image processing to be able to recover the maximum amount of structure information this vibration was included in the transfer function in the form of a Debye-Waller temperature factor. With the above considerations the total transfer function H(k) is:

Fourier transforms of dependent variables are denoted by capital letters. Fast Fourier transform (FFT) computer algorithms are generally available and make the numerical computation of Fourier transforms feasible [20-23]. Eqs. (4) and (5) may be simplified by the use of the Fourier convolution theorem. With the above notation the recorded image intensity and its Fourier transform are:

g(x) = h(x) ® fo(X) + [h(x) ® fo(x)]* -~n(x)

rrXk2 Re[x(k)] = (1 + eo k2) [~Cs(1 - eo kz) X2k2 - All , (65) Im[×(k)] =

[rrXq°k(QXZk2 - Af)]2 + ~(~rXA°k2)2 (1 + eo k2) (6c)

(8a)

= 2 Re (h(x) ® fo(x)} + n(x),

(8b)

G(k)=H(k)Fo(k)+H*(-k)F~(-k)+N(k)

(8c)

( ® represents convolution), where

fo(X)=~ue/s-~i.¢~s (withB=qo=Ao=×=O) (9a) X

H(k) = i(1 + eok2) -1/2 exp (-ix(k)] exp(-Bk2[4), (6a)

483

= l~t.c

12 ~

.

exp(2niq, ru) }

exp(2rrik • x/M) dk

(9b)

Fo(k) = [qJ~c[ 2 M~2{ ~ fu(a) exp(Zniq . ru) } . (9c) From eq. (8) it can be seen that because h(x) and H(k) are complex, parts of the real and imaginary

where Cs = coefficient of spherical aberration, Af = defocus value (varies with azimuthal angle for nonzero astigmatism), A o = standard deviation of defocus spread cause by instrumental instabilities and chromatic aberration, Lq o = illumination angle determining lateral coherence of incident electron beam, eo = (n3qoAo) a, and B = temperature coefficient. The ideal unaberrated image, fo(x), and its twodimensional Fourier transform (in the image plane), Fo(k) are related as

parts of the spectrum of the ideal image, fo(X), are intermingled (by h(x)) and the real part of the result is recorded. The ideal image fo(x) which is to be recovered is the interference pattern between the unscattered wave and the scattered wave. Because the unscattered wave does not differ appreciably from the incident wave which is constant across the specimen area the ideal image is essentially the complex scattering amplitude of the specimen, fo(X). I f B = qo = Ao = 0, then Im [x(k)] = 0, and eq. (8c) takes the familiar form [15]:

fo(x/M) = FT - I {Fo(k)}

G(k) = FR(k) sin [x(k)] + Fi(k) cos [×(k)] + N(k),

~fFo(k)exp(2rrik. x/M)dk,

(10a) (7a)

where FR(k) = 2FT{Re[fo(x)l} = Fo(k)+ F o ( - k ) ,

Fo(k ) = FT ~fo(x[M)}

(10b)

Fl(k) =-- 2 F T {Im[fo(x)] } = i {Fo(k) - F o ( - k ) } . (lOc)

- f fo(x/M) exp(-2rrik • x/M) dx/M .

(7b)

A defocus series of m micrographs of a given speci-

EJ. Kirkland et aL / Digital reconstnwtion of phase contrast images

484

men may be represented by (compare with eq. (8c)):

sin × l (k)-

Gu(k ) = Hul(k ) F l ( k ) + Hu2(k) F2(k ) + N . ( k ) ,(1 la)

sin ×2 (k)

[cos I (k) !

where la runs from 1 to m, or H21 --,l

2

H22 F

--

HR(k ) =

t

~'2

'

-

Li

r~

Hm l H m ;

×m(k

N2[ +

•

(I 1 b) " [

•

_Nrr]J

or

G(k) = H(k) F(k) + N(k),

cosm(*_

Henceforth, further explicit reference to the electron optical magnification, M will be dropped. The vector and matrix notation introduced above greatly condenses the representation of the hnear components 9 f a defocus series. Any linear algebraic treatment of a defocus series such as is presented in the next section is greatly simplified, as will be seen.

3. Image processing

Hul(k)=Hu(k),

(Xld)

Hu2(k) = H ~ ( - k ) ,

(1 le)

F l (k) = Fo(k),

(11 f')

F2(k) = F o ( - k ) .

(1 lg)

All of the quantities in eq. (11), except k, are treated as complex quantities. The subscript index/J in eq. (11 a)labels the different micrographs in the defocus series, m is the number of input micrographs. F(k) is a two-dimensional image vector. G(k) and N(k)are vectors of dimension m, each coordinate of which represents one micrograph. H(k) is an m X 2 matrix representing the transfer function of the series. Sansserif characters (H) will denote matrices. If either Fg(k) or Fi(k) (see eq. (10)) is not present, then eqs. (11) reduce to:

Gu (k) = Re [Hu (k)] F R(k) + N u (k),

(12a)

Gu(k ) = Im [Hu(k)] Fl(k) + Nu(k) .

(12b)

And in particular for perfectly coherent imaging Ao = qo = 0, no thermal vibration (B = 0) and Fi(k ) or FR(k ) absent:

G(k) = HR(k ) FR(k ) + N ( k ) ,

(12c)

or

=

, (12e,f)

(11 c)

where:

G(k)

.

I cos2 (k) I" Hl(k ) = t.

Hi(k ) F l ( k ) + N ( k ) ,

02d)

where H R (k) and Hx(k) are m-dimensional vectors;

The purpose of digital image reconstruction discussed here is to determine the ideal complex image intensity distribution fo(x), eqs. (9), from a series of micrographs taken at different defocus. The influence of the transfer function (eqs. (5b) and (6)) must be deconvoluted while at the same time maximizing the noise rejection• The possible benefits of such a procedure are increased resolution, improved SIN and/ or extraction of additional information about the spec. imen. Because of the feasibility of finding the Fourier transform (and its inverse) of an arbitrary image by numerical computation it is convenient to deal with the Fourier transforms of the images. In Fourier space the image model, eq.(11), is a simple linear algebraic equation. This transformation greatly simplifies the following mathematics. Image processing will be discussed in Fourier space only. It is assumed that the original micrographs of the defocus series have been inputted to the computer and numerically Fourier transformed. After image reconstruction has been conducted in Fourier space, the result is assumed to be numerically inverse Fourier transformed and outputted. Starting with the image model of the defocus series, eqn. (11), the OLS (ordinary least squares) method [11,15,16] solves for F(k) by multiplying eq. (11) by the pseudo-inverse matrix of H (k). This solution will be denoted by/~OLs(k) and is given by: /70Ls(k) = (HtH)

-1

HtG

(13a)

E.J. Kirklandet aL/ Digitalreconstruction of phase contrast images = (HYH)-' HY(HF + N)

(13b)

=F+ (HtH)-IHPN.

(13c)

The symbol/4 t denotes the hermetian adjoint of H formed by the combined operations of transpose and complex conjugation. The second term in eq. (13c)has the potential to dominate the OLS solution. In regions where the transfer functions H are small in magnitude (for instance, where the defocus-dependent attenuation envelopes dominate) the noise terms, N are multipied by large numbers. If low SIN micrographs are to be processed, N and F are comparable in magnitude. The second term then dominates the OLS solution and FOLS is large and meaningless in these regions. Also, there is the inconvenience of dealing with the regions at or near the singularities of ( H t H ) - I , as noted by Frank [ 11 ]. Near these singularities numerical roundoff error due to finite precision arithmetic in the computer is a major problem. The question of what is meant by "near the singularity" is itself ambiguous (in a mathematical sense). The operation represented in eq. (13a) must be performed at each point in Fourier or reciprocal space. Because the totality of these results must then be inverse Fourier transformed, an error at any point in Fourier space has the potential of dominating the entire area of the real space image. Something must be done to take into account the singularities and the possibly large perturbations N(k). Because N(k) is a random variable and only some of its statistical properties can be estimated, it is necessary to treat image processing in a statistical sense. /5(k) will now represent an optimal estimator, in a statistical sense of F(k). There are various criteria that can be used to judge the performance of any given estimator [17]. The simplest of these to use and understand is the minimum mean squared error (MMSE) criterion which has been adop.ted here. It requires that

f

fir(x)-?(x)':4

485

possible measurements of the defocus series. The signal F and the noise N are treated as uncorrelated random variables.Nis assumed to have a mean value of 0. The simplest form of the optimal estimator that restricts the additional required information to something that is realizable is a generalization of the OLS solution of the form:

f = Q(HtH)-I HtG .

(15)

Q is a single valued scalar function of k that must be found from eq. (14). Because the integrand in eq. (14) has a particularly simple form, it can be easily solved using the calculus of variations;

~e/OQ = O,

Oe/~Q*= 0 .

(16)

The first of these yields a solution for Q* while the second yields a solution for Q. Solving for Q from eqs. (14)and (15):

aQ* (IF- Q ( H t H ) - I H t ( H F + N ) I 2 ) = O ,

(17a)

aQ, (1(1 - Q ) F - Q ( H t H ) - I H t N I 2) = 0 .

(lVb)

Because the signal F and noise N are assumed to be uncorrelated, a aQ* (11 - QI2(IFI2) + IQI2(I(HtH ) - I H t N I 2 ) } = 0 . (17c) The noise in one micrograph is assumed to be uncorrelated with the noise in any other micrograph and the noise in each micrograph is assumed to have the same power spectrum:

(NuN v) = 6m,(lNu 12), (INul2)=(lNvl2)=(INl2/rn),

(18a)

la=/=v.

(18b)

m is the number of input micrographs. 6uv is the Kronecker delta. For a general matrix M this implies that

(N'tMN) = (INI 2) T r [ M ] / m .

(19)

Tr represents the trace or spur operator. Eq. (17c) now takes the form: = minimum.

(14) -

( ..- ) denotes statistical averaging over an ensemble of

-

aQ*

{[1 - QI2(IFI 2)

E.J. Kirkland et al. /Digital reconstruction of'phase contrast images

486

+ IQI2(INI2/m) Tr[H[(HtH) -l ] t(HtH)-I H t] }

= 0.

(20a)

Using various trace and matrix identities, this reduces to a aQ*

{I1-QI2 +IQI2rTTr[(HtH)-I]}=O'

(20b)

where r/= (INI 2)/m(IFI 2 ) is the average noise-to-signal power ratio for one micrograph in reciprocal space for an identity transfer function ( H = I ). Taking the indicated derivatives in eq. (20b) gives Q= 1/{1

+ r/Tr[(HtH)-X] }.

(21)

Finally the MMSE solution is:

(HtH)-'H t 15 = 1 +r~Tr[(H+H) -1] G

(22a)

= {1 + rl T r [ ( H t H ) -1 ] }-1 [F + ( H t H ) - ' HtN] . (22b) In the computer this operation must be performed at each point of the Fourier transform of the input images and the results inverse transformed to obtain an estimate of the real space ideal unaberrated complex image fo (x). After combination of several independent measurements of Gu using eq. (22), the effective transfer function of the resultant estimate of F, ,~ is a positive real scalar function:

Herr = {1 + r/Tr[(HtH )-l] }-, .

(23)

The functional form of this estimator (eq. (22a)) is remarkably similar to the well known Wiener inverse •ter which is the MMSE solution for an H of dimension 1 X 1 [17,20-22]. The second term in the denominator, rffr[(/-#H) - l ] is a measure of the relative significance of the noise power (INI 2) and the transmitted signal power ([HFI 2) with allowances for the singularities in (HtH) -1 and the statistical averaging of m independent measurements of Gu . In regions where the recorded signals G have no information about the desired signal F (i.e. (IN[ 2) > > (I HFI2)) it is impossible to say anything about the signal F and the optimal estimator F is approximately zero. In regions where the transmitted signal is strong (i.e. (IHFI2) > > (INI2)) the optimal estimator reduces to the OLS solution/~ ~ F. The form of the estimator

as given in eq. (22) has been chosen to join these two regions in an optimal sense (eq. (14)). The signal-tonoise ratio SIN ~ 1/7/of the original input micrographs determines how close to the singularities and how far out into the attenuation envelopes to take information. The higher the initial SIN the more information that may be recovered. In particular in the limit of infinite SIN the MMSE estimator and the OLS solution are identical and complete recovery of the original information F is still possible in principle. The above considerations lead to the conclusion that the MMSE estimator is more efficient in terms of noise rejection that the OLS solution. The MMSE estimator, eq. (22), is the solution to the general linear inverse problem corrupted by noise as characterized by eq. (11) (i.e. H can be of any dimension). However, for the particular problem considered here:

I;l

12 L~v

/a

)]-,

+n(~] IH.,12+ ~] IH.212 #

(24)

Because the denominator is always denominator/> r~(~] IHu112+ ~ IHu212) / /a \u

(25)

(by Schwartz's inequality), the singularity suffered by the OLS solution [11,15] has been removed in the MMSE estimator. Also the effective transfer function eq. (23) reduces to:

Herr=~ IHv1121H~212- I S H~IH.2* 12 /.w

X[~

/~

IH.,121H~212- I S

Hj,H~212

)]-,

+n ( ~ IH.112+ ~ 'H,a 12

(26)

It is interesting to note that if H has dimensions m X 1 (i.e. a vector H = H ) the MMSE estimator (eq.

ILJ. Kirkland et al. / Digital reconstruction o f phase contrast images

(22)) reduces to

~:1~.a n + IHI 2 '

(27a)

which is Schiske's MMSE (reduced) Solution [ 18,19]. This form of the MMSE estimator is appropriate for a defocus series with image model eq. (12) in which only the real or imaginary image is present. This form of image reconstruction has been studied by Saxton et al. [30], Uyeda et al. [31] and Kirkland and Siegel [19]. Furthermore if H has dimensions 1 X 1 (i.e. a scalar H =H ) the MMSE estimator reduces to F=

H*G

(27b)

rt + IHI 2 '

which is the well known Wiener inverse filter. This form of the MMSE estimator is applicable to single micrographs with image model eq. (12a). For this reason the MMSE estimator eq. (22) can be thought of as a generalized Wiener inverse filter. It is hoped that the performance of eq. (22) will be as good as that evidenced in [20-22] for the Wiener filter. The image reconstructed using eqs. (22) and (26) can be checked for consistency by recalculating a defocus series from the reconstructed image using the parameters of the original micrographs and comparing it to the original defocus series. This recalculated defocus series is found from eqs. (22), (26) and Greta, : H F',

(28a)

and should be very similar to the original defocus series except for an increase in signal-to-noise ratio (S/N). In fact by numerically varying the input parameters of one micrograph and numerically searching for -

flG.,recm(k) -

Gu(k)l 2 dk = minimum ,

(28b)

the input parameters (such as Af) of the original series may be refined. The OLS solution (eq. (13)) may also be phrased as the solution to a less important MMSE criterion. FOES as given in eq. (13) is the solution of (fig

- H/~OLS 12 dk) = minimum

found by varying FOE s instead of Q* as in eqs. (15) through (22). This criterion (eq. (29)) is less restrictive than eq. (14) because it ignores the regions where the transfer functions H are small and does not take into account the possibilities of singularities in (H t H ) - l . The sensitivity to errors in the specification of the transfer functions H (k) and alignment of input images will not be discussed in depth here. However, the error sensitivity of similar methods have already been considered. Frank [11 ] discussed analytically some effects of small errors in Af i n the OLS method. By computer simulation using noise-free simulated images Welles [32-34] found that the OLS method was extremely sensitive to small errors in alignment of the input micrographs and specification of H. The reduced form of the MMSE estimator given in eq. (27a) has been studied analytically [19] with similar results. Because of its essential similarity to the above methods, the MMSE estimator (eq. (22)) is assumed to have the same accuracy requirements as those found for the reduced form of the MMSE estimator [19]. In particular the required accuracy of alignment between different input micrographs is identical to that discussed in ref. [19]. Starting with eq. (24) of this paper and eq. (19d) of ref. [19], the alignment accuracy requirement for the inaage reconstruction method presented here may be derived in a manner identical to that in [19] with an identical result. If 15xl represents a misalignment of one micrograph with respect to another, then laxlmax < ~ Amin (also eq. (21b) of ref. [19]),

(30a)

where Amin is the desired resolution of the reconstructed image. Also the maximum total error in Re(x) (eq. 6b) was found to be limited as

f lgr.ca~(X) g~(x)? dx =

487

(29)

laxl <~ n / 3 .

(30b)

Rose [35] and Downing [36] have recently pointed out that the imaginary part of the ideal image fo(x) (eqs. (7)-(9)) due to kinematical elastic scattering is a small signal, in which case neglect of the quadratic image term I ffs 12 (eqs. (1)-(4)) may not be appropriate. Welles [32,34] and Welles et al. [33] have shown that Schiske's OLS method [11,15] can distinguish the imaginary image in the presence of the

488

E.J. K#'kland et al. /Digital reconstruction o f phase contrast images

quadratic term by processing a noise-free simulated defocus series including I Cs 12. The MMSE method of image reconstruction introduced here has also been applied to a noise-free simulated defocus series by Kirkland [37]. The reconstructed image was in excellent quantitative agreement with the original image wave convoluted with the effective transfer function (~bu + ~bs) ® heft (eqs. (1) and (23)). The total rlns error was found to be less than 0.68% in the real part of the image and less than 3.10% in the imaginary part of the image. Because the human eye can distinguish only 2 0 - 3 0 grey levels this error is completely negligible. This rejection of the quadratic term may be understood from an examination of eqs. (6) and (24). The relevant quantities obtained from the defocus series that are used in the reconstruction algorithm (eqs. (22) and (24)) are EHul Gu and EHu2G u. Because H~I and H~2 have a strong oscillatory nature any part of Gu that is not correlated with Hul or H~2 (such as the Fourier transform of n(x) or I¢s 12) will average to zero if a large number of input micrographs are used. Therefore neglect of the quadratic term in eqs. (1)-(4) is entirely appropriate for the particular method of image processing discussed above.

4. Input images The input micrographs were taken on the Kyoto 500 kV high resolution microscope [38]. A tilted thin crystalline specimen of hexadecach.lorophythalocyanato-copper (II) (abbreviated CuCII6PC) at room temperature with known structure was used. Fig. 2 shows the input defocus series of five micrographs after some initial photographic processing (explained below). Also shown in fig. 2 is a model of the twodimensional structure of the specimen projected onto the image plane. The structure of this specimen is thought to be primarily two-dimensional. The crystal structure is monoclinic with C2/c symmetry and lattice parameters [39]: a =19.62A,

b =26.04A,

c = 3 . 7 6 A , /3 =116.5 ° .

It consists of planar molecules arranged in planes as shown in fig. 2 (top or bottom view, c-axis perpendicular to plane of paper). These planes of molecules

are stacked one plane on top of the other. When the c-axis of the crystal is aligned with optical axis of the microscope, corresponding atoms of different molecules in different planes line up parallel to the optical axis of the microscope. The resulting images are the superposition of the images of all of the atomic layer (planes of molecules) of the crystal at slightly different defocus values, and if the specimen is thin enough (linear-kinematic region - eq. (3)) reproduce the two-dimensional structure of these planar molecules as shown in the model in fig. 2.

4.1. Specirnen thickness effects To satisfy the linear-kinematic conditions, eq. (3), the specimen thickness was kept at approximately 10 atomic layers ( ' 3 8 .~). The maximum thickness tmax for which the linear-kinematic scattering approximation, eq. (4), is valid at 500 keV was found to be 30 atomic layers by a multi-slice calculation [40,41 ] considering dynamical elastic scattering. At an incident electron energy of 100 keV this upper bound would be even less (t <~ 20 atomic layers). The linear-kinematic conditions (eq. (3)) including some inelastic scattering effects may also be checked in a simpler manner by use of the approximate analytical expressions for the elastic and inelastic scattering cross sections: oe-

1.5 X 10-4Z3/2[ 0.23Z\ 2 ~ ~ 1 - 1 3 - - - ~ } ( A )-+30%, (31a)

oi ~ 2 0 o e / Z

(3 lb)

,

as given by Langmore, Wall and Isaacson [42] and Lenz [43]. Z is the atomic number of the scatterer. /3 = o/c is the velocity of the incident electrons relative to the speed of light (/3 = 0.86 for 500 kV electrons). For this specimen

POe = ~

PuOeta= 1.62 X 10 -3 , po T = ~

+ oiu ) = 4.57 X 10 -a

pu(Oeu (32a)

(in dimensionless units where the thickness t is measured in the number of atomic layers). Eq. (3) now becomes, using eq. (31):

1

1@"12

1¢4,¢ 12

1 - exp(--POTt)

E.J. Kirkland et al.

/

Digital reconstntction o f phase contrast images

(a)

489

(b)

(c)

(d) 0 0

0 0

o.':.o

o-:.o

0 oe eo 0

O o o o oa ~ _ o . o m O ~°o°e''lw-°°o °~'~,

"0o

0~o

0 oe eo 0

0 0

O . o a o_ "--°.o oOoO n °o°°-l'-°°o°~

o~o

0 ~00°%0-0 O. leO ^ *o -

O-

0.~[*0

o% o.... l . . . . o o'Fo •

~

o • • o

u

•

,-, I,-, Jr'el v~v o °,.,o ° °0°.~. . . ° ° oo ° 0 Cu ~'~' O0-Fo 0 v O0 £1'-~.I/~CI o

•~-'.

o

c, , . ~ ,

o:.*o

o:•, • . ' ~ . - e 0 - . ~ ' 2 C 0 ~e io O o0%

-o"o

(e)

Q

ci

0

-~1

CI-~-C, ,

loA

,

~,, o, u

~,%~,

(f)

Fig. 2. Original micrographs taken on the K y o t o 500 keV electron microscope. (a) A f = 1619 A (K = 9); (b) ~ f = 1421 A (K = 7); (c) ~ f = 1191 A (K = 5); (d) ~ f = 903 A (K = 3); (e) ~ f = 461 A (K = 1); (f) model o f the specimen structure as projected into the

image plane. All defocus values are -+20 A, and the scale of all images is shown in (r) with a 10 A bar. 0.0447 < ( S / N ) -1 ~ 0.1 , [~s 12 [~u 12

1 -- e x p ( p o e t ) exp(--po T t)

(3a)

0.0168 < ( S / N ) - l ~ 0.1 . (3b)

The anticipated S I N is ~10 so that these linear-kinematic conditions (3a) and (3b) are satisfied. Eq. Oa) may also be inverted to give: tma x < < 23 ± 7 atomic layers.

(32b)

This is a more stringent requirement than the above

mentioned multi-slice calculation because first-order inelastic effects have also been included. At 100 kV similar considerations require a crystal thickness t < < 9 ± 3 atomic layer. (If inelastic scattering is ignored then t < < 27 ± 8, 65 + 20 atomic layers for 100 kV and 500 kV, respectively, in agreement with refs. [40,41] .) Therefore the specimen thickness used here (10 atomic layers) clearly satisfies the linearkinematic conditions. In so far as kinematical scattering theory is applicable the effects of the three-dimensional extent of

E.J. Kirk~andet al. / Digitalreconstructionof phasecontrast images

490

the specimen on the phase of the scattered wave may also be calculated analytically. Kinematical theory, however, cannot be expected to give accurate quantitative results [44,45], but can reveal the salient features of the scattering process. The multi-slice calculations cited above [40,41] are the most accurate means of calculating three-dimensional effects. Hirsch et al. [44], Hoppe [13], Spence et al. [45] and Zeitler [46] have illustrated analytically some kinematical effects of specimen thickness on resolution, although little work has been done using complex scattering amplitudes. The specimen used here has a tilt of 8.8 A/molecule (26.5 ° to optical axis) and a thickness of ~38 A. Specimen thickness is assumed to be the dominant effect and the flit is ignored. Allowing for complex scattering amplitudes the Fourier transform of the ideal image is (see fig. 1): F o ( k ) = I ¢ ' ~ 12 ~X

X ~. = ICmcl2 m-2

~u fu(a)exp(2rriq " ru )

(33a)

[u(a ) exp[2rri(k.x u + IqzlZu)] (33b)

=

I~incl 5 ~X ~uf . ( c ~ ) e x p

rri

.x

u+~-~zu

.

atoms,

NI2

2 NI2 cos(rrk2 ~gu)

sin(~ 7rk2 kNc) = N sin(½rrk2Xc)

(36)

= kinematical thickness modulation factor - 1 < T(k) < 1. The kinematical thickness modulation factor r(k) for various thicknesses of this specimen and an imaging electron energy of 500 keV (X ~ 0.0142 A) is shown in fig. 3. For a resolution of 1.25 A and a thickness of 10 atomic layers r(k) falls to only ~0.95 which is negligible compared to other sources of image degradation. Therefore, the recorded image (eq. (9)) is essen tially the image due to one plane of atoms multiplied by the number of atomic planes. The most important point brought out by this calculation is that the "curvature of the Ewald sphere" problem encountered by Frank [11,12] has effectively vanished from this specimen. Frank studied stained DNA supported on thin carbon films. Because an externally supported specimen is not symmetric in z the factor expQd k2~.u) in

(33c) This specimen is symmetric in z (coordinate parallel to c-axis of the crystal and the optical axis of the microscope) and is composed of N atomic layers. Therefore:

4 ATOMIC LAYERS

1.0

A , OM,C

Fo(k) = 21 ffinc 15 M-3

0=E2~

X { ~ fu(°O exp(27rik " xu)}xy

X-- O.O'42 A

LAYERS

=

z

NF~(k) r ( k ) ,

(34a) (34b)

where

Fo(k) =

I ¢'i.,: 12 ~X

~u fu(a ) exp(2rrik, xu)

\

zn-

A=l/k

~ J

16 ATOMIC~

--

0.5

(35)

= scattering amplitude from one x - y plane of

I

8

4

2

I

I

I

I

I

I

I

0.5

in ,~, 1.5 L25 I

I

I

I I

k : I/-A,. in

I

~-I

I I

P

I

1.0

Fig. 3. Modulation factor due to the finite extent of the specimen in the z direction (parallel to the optical axis of the microscope) and the curvature of the Ewald sphere(fig. 1) calculated within the kinematical scattering theory for a symmetrical (in z) specimen versus spatial frequency k. Electron wavelength h, and lattice spacing c in the z direction are shown.

E.J. Kirkland et aL /Digital reconstntction o f phase contrast hnages

eq. (33) causes the real and imaginary images to be mixed and any possibility of heavy/light atom discrimination is diminished. Because the specimen used here is symmetric in z the real and imaginary images remain unmixed. In fact the thickness of the specimen advantageously increases the total signal-to-noise ratio with negligible side effects. Therefore, the reconstructed image should be a reasonably accurate rendition of the complex wave scattered from a single layer of CuC116Pc molecules.

4.2. Radiation damage Although this specimen is one to two orders of magnitude more resistant to radiation damage than other organic compounds [47-49], it is still very sensitive to radiation damage. Each micrograph must be taken with a low radiation dose and hence low SIN. A total dose of less than 2 C/cm 2 was used to expose a 17-micrograph defocus series (the critical dose is ~50 C/cm2). The low DQE of electron plates at 5,00 kV also severely limits SIN. Even though all 10 atomic layers contributed (at slightly different Af) to the image (increasing SIN by ~10 as compared to a single molecule) there was still not enough SIN present in the original micrographs to make atomic resolution possible. To increase the effective SIN of the input micrographs, each micrograph was averaged over 5 unit cells by photographic superposition using the Cu sites as reference points and translating along the b-axis to further increase SIN by "x/5. This transitional averaging procedure requires prior knowledge of the periodicity of the specimen and would not be applicable to a completely unknown (no observable periodicity in original micrographs) or amorphous specimen. To further take advantage of the symmetries of this specimen, each image was reflection symmetry averaged about the center line of reflection symmetry (again by photographic superposition) to further increase the effective SIN of the input micrographs by a factor of~/2. Only parts of each of each micrograph with equal defocus were superposed. This reflection symmetry averaging also tends to enhance the noise along the center line of reflection symmetry. However, note that the image of the center molecule appears to be identical (within SIN limits) to the four corner molecules (see fig. 2). Because this center line noise enhancement should

491

only affect the center molecule and not the four corner molecules it appears that center line noise is not a problem. There are no artificially induced image artifacts. Therefore the images used as inputs for image reconstruction have an effective SIN approximately 10x/10 = 32 times larger than a conventional micrograph of a single molecule.

4.3. Electron opticalparameters Numerical values of various required electron optical parameters are tabulated in table I and were measured as described below. Incident electron wavelength k was measured from Kikuchi pattern spacing as first described by Uyeda et al. [50]. Overall magnification including electron optical and photographic effects was obtained from the projected lattice parameters b and a sin/3 using the copper atom sites as reference points. Spherical aberration Cs was inferred from the defocus dependence of the displacement between a bright field image and the corresponding dark field images due to two different Bragg beams with the objective aperture removed in a manner similar to that described by Bu'dinger and Glaeser [51 ]. The half width of the defocus spread Ao was estimated from the measured instabilities of the high voltage and lens current power supplies, the energy spread of the source and the chromatic aberration constant supplied by the manufacturer of the microscope (JEOL). The illumination semi-angle a e was measured from the broadening of diffraction rings in a SAD pattern of gold particles (also done by JEOL). This value includes both geo-

Table 1 Parameter

Value

h M

0.0142 ± 0.0001 A 1.483 X 107 ± 1.5% (0.2697 A/pixel) 4 6 1 , 9 0 3 , 1191, 1421, 1619 ± 20 A For K = 1, 3, 5, 7, 9 respectively (see eq.

~f G c~ A0 ~C amax

(37)) 1.06 -+0,09 mm 1.6 mm ± 10% 75 ± 25 A 0.6 ± 0.2 mrad 14.0 ± 0.5 mrad (objective aperture limits resolution to 1 A)

E.J. Kirkland et al. / Digitalreconstruction of phase contrast images

492

metrical and objective lens prefield factors. Defocus Afwas chosen with the aid of the tilt of the crystal. Seventeen successive micrographs were taken. Each micrograph has a defocus that varies continuously from the top to the bottom of the micrograph ~8.75 A/molecule-along the a-axis due to the tilt of the crystal. Between micrograph exposures the objective lens current was changed by a small amount. Because the specimen is translationally symmetric (i.e. it is crystalline), the top of one micrograph can be visually matched in a defocus to the bottom of the next. In this way the change in defocus can be found continuously from the first micrograph to the last. A reference value of Afwas set by observing the position of the "best focus" image and assigning this as "Scherzer focus". The appearance of the "best focus" image is known from multi-slice calculations [40,41]. Other defocus values were chosen by moving a prescribed number of unit cells ( 6 A f ~ 8.75 A/molecule) away from the "best focus" position. Optimum defocus, Af was chosen from: Af= [(2K - 0.59) CsX] 1/2

K = 1 , 2 , 3 . . . . . (37)

as described by Eisenhandler and Siegel [52]. K = 1 corresponds to the "best focus" or "Scherzer focus" position. These defocus values produce wide flat bands in sin × for perfectly coherent illumination. These bands pass large amounts of information about the real part of the image and are insensitive to small errors in the magnification [19]. K also serves as a dimensionless label of Af. The astigmatism of the objective lens was accurately compensated before exposure of the defocus series by observing the ellipticity of the rings in an optical diffraction pattern of an electron micrograph of a thin carbon film [53]. Defocus can be assumed to be independent of azimuthal angle. The measurement errors in the electron optical parameters in table 1 effectively limit the maximum resolution that can be recovered by image processing. By comparison of the errors listed in table 1 with eq. (30b) and ref. [19] the maximum resolution (smallest resolvable distance) is estimated to be of order: Ami n (electron optical parameter errors) ~ 0.5(CsX3) 1/4 1.17 A .

(38)

4.4. Transfer functions Using the above mentioned independently measured electron optical parameters in table 1, the instrumental part of the transfer function (eq. (6) with B = 0) is shown graphically in fig. 4a. This part of the transfer function is determined solely by the microscope. The real and imaginary parts of the transfer function H(k, B = 0) are plotted separately for K = 1,3, 5, 7, 9 (see eq. (37)) versus the normalized spatial frequency (CsX3)l/41kl = 2.35 A/A. The part of the transfer function due to thermal vibration (the Debye-Waller factor exp(-Bk2/4) in eq. 6a) is shown in fig. 4b. Thermal vibration is normally included in the scattering amplitude of the specimen and is not part of the instrumental transfer function. However, it is an undesirable effect that blurs the observed image of the specimen structure (the high spatial frequencies are attenuated). To account for the loss of high frequency signal and allow some correction of this effect, thermal vibration has been included as part of the total transfer function used in the image processing algorithm (eqs. (22) and (24)). A typical value of B ~ 4 A 2 for organic crystals at room temperature as quoted by Vainshtein [54] has been used here. For comparison the Debye-Waller factor with a typical value o f B ~ 0.5 A 2 for inorganic crystals at room temperature [54] has also been plotted in fig. 4b. It can be seen from fig. 4b that at the 1 A resolution level the Debye-Waller factor is a major factor for organic structures but negligible for inorganic structures. Because the specimen used here is organic it is necessary to include thermal vibration effects.

4.5. Alignment The requirement of very accurate alignment between input micrographs found in ref. [19] also exists for the image reconstruction method presented here. To achieve this high accuracy each micrograph was carefully printed (enlarged) onto large sheet f'dm (10.16 × 12.7 cm) with all five copper atom sites accurately positioned with respect to two edges of t h e film. These two film edges were then used to position the film on an Optronics P-1700 photoscanner for scanning and digitization. The final separation between the two Cu atomic sites on the b side (longest side of rectangle) of the unit cell was ~3.86 cm. Ea.ch

i

W

K°I

K'I

"1

K°3

~

K'3

f-,,=

K'5

K°5

I

ol

K'7

•

Ko7

V 'vv -I

K'9

1.01

b

~.

~ Fig. 4. (a) Instrumental part of the transfer function (determined by the microscope) versus normalized spatial frequency (Cs h3) 1/4 Ikl = 2.35 A/A (eq. (6)). The real part of the transfer function Re H, and the imaginary part of the transfer function Im H are shown. K is a dimensionless label of defocus (eq. (37)). Other parameters are tabulated in table 1. (b) Debye-Waller temperature factor part of the transfer function due to thermal vibration of the atoms in the specimen versus spatial frequency (eq. (6)). For the specimen used here B = 4.0 A 2 (organic specimen). A curve for B = 0.5 A 2 (inorganic specimen) is also shown for comparison.

0.5

T

e~

? 4I J I I

A

t

.

)

o,\

,oRo.ccYo, A = I/k in 2I ,p ,.p I I I I I 0.5

k = I/-A..

in ~-I

I

,I I

1.0

I

D ,-

494

E.J. Kirkland et al. / Digital reconsmtction of phase contrast hnages

of the outer Cu sites was aligned to approximately +0.5 mm (0.34 A) (rms. stnd. dev.). Because of the high redundancy and large separation of the alignment points (Cu sites) the alignment accuracy of the central molecule is approximately +0.25 mm. From eq. (30a) thc resolution limit set by alignment errors is: 16xlm,x ~< 0.17 A ,

(39a)

Amin(alignment) > 616Xlmax ~ 1.01 A .

(39b)

This alignment accuracy is better than the "Scherzer" resolution of the input micrographs [0.66 (Csk3) l/a 1.55 A] and could only be attained by using many redundant strong scattering sites (5 Cu atomic sites) and requires some specific knowledge of the specimen.

cessed images. The Versatec 1200A can also be used as a line printer or a raster scan plotter (e.g. figs. 4a and 4b are Versatec outputs). The film writing portion of the P-1700 photoscanner was used to produce pemaanent negatives of all outputs shown here. All programming was done in either Fortran IV or machine language. The overall image reconstruction procedure including Fourier transforms, input/output, etc., was segmented into smaller subprograms and run under the DOS/BATCH V9 operating system distributed by DEC. The overlay structure initiated by Welles [32] was modified to perform the new reconstruction method presented here (see eqs. (22), (24), (27) and (28)) and to accommodate recently added hardware. As a measure of the computational speed of this system, a 128 × 128 complex to complex fast Fourier transform (FFT) takes approximately 12 min of CPU time.

5. Experimental image processing 5.2. Measurement o f the noise to signal power ratio 5.1. Computer system

The input images (fig. 2) described in the previous section were digitized on an Optronics P-1700 scanning microdensitometer with a 100 × 100/am square sampling aperture (25 and 50/am apertures are also available). A total area of 512 × 512 pixels (sampled points) on each of the five input images was scanned and reduced to 128 × 128 pixels by nearest neighbor averaging (400 × 400/am effective pixel size = 0.270,8,/ pixel) during the scan. Each 128 X 128 image was stored in a separate area on a magnetic disk. Numerical processing of the digitized images was done on a Digital Equipment Corp. (DEC) PDP11/20 computer with 56 Kbytes of CPU memory and auxiliary floating point hardware manufactured by Floating Point Systems Inc. (FPS model FP-03). A 2.5 Mbyte hard magnetic disk (DEC RK-03) was used as an image storage working area. Another 2.5 Mbyte disk was used for program storage. A Pertec T5540 magnetic tape drive (1600 bpi, PE, 37.5 ips)was also available for bulk data storage although it was not used for this work. A Grinnell Systems GMR-27 refreshed CRT image display and a Versatec 1200A (200 dot/inch) electrostatic matrix printer/plotter with specialized greyscale production software were used for rapid intermediate visual inspection of the input and pro-

The noise-to-signal power ratio, r/(k) of the input defocus series was determined from the power spectra of the original images. As part of the image reconstruction procedure the Fourier transform of each digitized image was calculated numerically and stored on the magnetic disk for future use. The power spectrum of each image was calculated by averaging the square modulus of the Fourier transform of each image over azimuthal angle. The average power spectrum of the defocus series was calculated by summing the power spectra of all five input images. A log-log plot of the average power spectrum is shown in fig. 5. This power spectrum may be separated into two distinct regions. In the high frequency region ((Csk3) 1/4 Ikl > 2) the transfer functions (see figs. 4a and 4b) are zero and the averaged power spectrum is dominated by the power spectrum of the noise. In the low frequency region ((Csk3) 1/a Ik l < 2)the transfer functions (see figs. 4a and 4b) are large and the averaged power spectrum is essentially the sum of the power spectra of the noise and signal. The averaged power spectrum in both regions was found to obey a Ik1-2 power law, indicating that r/(k) is a constant independent of frequency. A value of r / ~ 0.1 was obtained by extrapolating the observed power law dependence from the high frequency region into the low frequency region. This observed Ik1-2 law has not been explicitly

E.J. Kirkland et ill. / Digital reconstruction o f phase contrast images i0 II

I

I

I

I I

I

i

I

I

I

i

I ~ I

AVERAGE \

POWER SPECTRUM

\

OF INPUT DEFOCUS SERIES

\\\

\\~

I0 '°I.t_ I..I.. I..1..

0

.../ 123 0

~

108-

c :

\

\

"\ \\

n~

2

~/~\(S+N)oc k ~,\\S/N~IO

-. 0

O3 r-, 107"

6

\

-.

•

""\\~

N ,.,. k- z

hi n-"

bA ,~ I0 ~

N O R M A L I Z E D SPATIAL FREQUENCY iO 5

,

,

,,

I

O. I

,

i

,

,

i

I

i

,

i I

I.O

(c, x3) ~ I~1 Fig. 5. Log-log plot of the average power spectrum of the input micrographs (defocus series - see fig. 2) versus the normalized spatial frequency (Cs h3) 1/41kl.

accounted for theoretically but would be consistent with spatially averaged white noise in the high frequency region (i.e., integration with respect to x when Fourier transformed and squared yields a Ik1-2 factor). The observed Ik 1-2 dependence of the signal (in the low frequency region) has not been included in the image reconstruction procedure (eqs. (22) and (24)), although some additional simple high frequency emphasis filtering has bee.n tried (see fig. 10). This power law dependence of the signal may be due to the transfer function of the electron recording plates used to take the original micrographs which has been ignored here because it is not known either experimentally or theoretically for 500 keV electrons.

5.3. Results The effective transfer function Heff(k) (eqs. (23) and (26)) is plotted in fig. 6 for various combinations of input micrographs and S/N ratios (~l/rT) versus the normalized spatial frequency (CsX3)1/4 Ikl. The input

495

micrographs are labeled with K, the dimensionless label of defocus (eq. (37)). The right-hand side of fig. 6 shows the effects of the SIN ratio on the resolution obtainable by the image reconstruction method discussed above, as applied to all five input micrographs. An absolute resolution is shown in angstrom units. In the limit of large SIN (S/N> 1000), a resolution approaching 1 A is possible while a low SIN (S/N < 1.0) does not permit any reconstruction at all. With an SIN ratio of 10 as measured above, a resolution of approximately 1.23 A may be obtained. The lefthand side of fig. 6 shows the effects of using different input micrographs and indicates that three or more input micrographs are required for a reliable reconstruction. Because there is a formal singularity at the origin for all possible combinations of input micrographs (i.e., H(0) = 1 regardless of defocus), the effective transfer function cuts off the low spatial frequencies. This is in fact desirable for high resolution studies and has the added advantage of attenuating the inelastic scattering effects which predominantly affects the low spatial frequencies (i.e., inelastic scattering occurs mainly at small scattering angles). The image reconstruction procedure (eqs. (22) and (24)) was applied to various combinations of the input micrographs (fig. 2) using the parameters given in table 1 and r/= 0.1. The results are shown in fig. 7. The image reconstructed from only two input micrographs (fig. 7a) is a quite reasonable result, and as more input micrographs are included in the reconstruction procedure the reconstructed image improves and asymptotically approaches the final best result (fig. 7d) with five input micrographs. This is consistent with the effective transfer function Hell(k) (eqs. (23) and (26), and fig. 6). Fig. 7a also indicates that image reconstruction is possible from as few as two input micrographs (this corresponds to solving for two unknowns in two variables), although three or more micrographs are required for a reliable reconstruction, as would be predicted from the effective transfer function (fig. 6). The effects of varying the value of the noise-tosignal power ratio (r/) inputted to the image reconstruction procedure are shown in fig. 8. Specifying 7/too large should increase the noise rejection at the cost of decreasing the recovered signal and thus reducing the resolution of the reconstructed image.

E.J. Kirkland et al. /Digital reconstruction of phase contrast images

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Specifying rt too small should allow noise to come through into the reconstructed image and may give spurious structure regardless o f the noise. These effects are observed experimentally in fig. 8. It appears that specifying a value slightly larger than the real value

results in a subjectively better image (i.e., the human eye and brain are less confused by slightly less resolution than by extraneous noise). Overall it appears that the MSSE image reconstruction procedure is not very sensitive to an exact value o f ft. One significant figure

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497

is sufficient. This insensitivity is advantageous because this parameter is oftentimes not measureable. As has been mentioned above the MMSE image reconstruction algorithm presented here reduces to Schiske's original method (re f. [15] and eq. (13)) in the limit of infinite SIN (~l/r/). As implemented here the MMSE image reconstruction procedure may be converted into Schiske's original method by setting r/= 0. The results of Schiske's original method are shown in fig. 8f. As is obvious, Schiske's original method does not work very well. In fact, when executed the program had one recoverable run time error (i.e., there was one division by zero, the results of which were ignored and program execution was continued by the operating system). The principal regions of failure for Schiske's original method are at the high spatial frequencies where the attenuation envelopes have totally cut off the signal (i.e., noise is multiplied by large values) and at the singularities of the matrix inversion (see eq. (13)). To check the validity of the image model (eqs. (4) and (8)) and the image reconstruction algorithm (eqs. (22) and (24)) and the precision of the programming, reaberrated images were calculated by applying the transfer function corresponding to each of the original micrographs to the reconstructed image as in eq. (28). The digitized original images are shown in fig. 9a (compare these to the original undigitized images in fig. 2). The reaberrated reconstructed images are shown in fig. 9b. Each of the images in fig. 9b was calculated from all five images in fig. 9a. It should be noted that in the calculation of the reaberrated reconstructed image (fig. 9b) the results were not constrained to be real. F I (k) and F2(k) (eq. (I 1)) were calculated as completely independent quantities at each point in Fourier space and were not constrained to obey Friedel's law (Fok) = F o ( - k ) ) . The mere fact that the reaberrated images (fig. 9b) are real is a good indicator that the linearized image model (eqs. (4) and (8)) is a good approximation. The reaberrated images should also show a greatly increased S/N, a loss of very low frequency information (and presumably a reduction of inelastic scattering effects) due to the low frequency cut-off of the effective transfer function (eqs. (23) and (26)) and presumably a loss of the quadratic terms I ¢s 12 and I ~bu 12 (eq. (2)). With these considerations in mind the general agreement is quite good. Fig. 9c is the difference between fig.

498

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9a and fig. 9b (the K = 1 micrograph appears to have a scale problem). The best results obtained are summarized in fig. 10. Some additional high pass filtering was applied to the real part of the image to increase the apparent resolution. Some low pass faltering was applied to the imaginary part of the image to decrease the high frequency noise. All five input pictures were used.

6. Discussion Recovery of the complete complex (both real and imaginary parts) phase contrast image of thin specimens has been shown to be possible by digital image processing of a low S I N (=10) (low radiation dose) defocus series of conventional transmission electron micrographs. This is equivalent to finding both the

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500

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imaginary Fig. 10. Best reconstructed image. All five input micrographs (fig. 2) were used ( a c = 1.0 mrad). The real part of the image has been high frequency emphasis filtered by multiplying by frequency in Fourier space. phase and amplitude of the scattered wave in X-ray diffraction structure determination and has many advantages in atomic structure determination. Specifically, the imaginary part of the reconstructed image shows greatly increased contrast between atoms of different atomic number facilitating "heavy/light" atom discrimination. The image reconstruction procedure also yields an overall increase in resolution by partially correcting some of the instrumental deficiencies of the electron microscope, and an overall increase in the average signal-to-noise ratio, S / N , by combining many sources of information in an opti-

mal fashion. These benefits have obvious advantages in high resolution atomic structure determination. Image reconstruction utilizes the variable phase retarding properties of the magnetic lenses of the microscope. By observing the changes in the recorded micrographs versus the changes in the spatial frequency-dependent phase retardation (calculated theoretically from measured instrumental parameters) in the imaging system of the microscope, both the phase and amplitude of the imaging electrons can be recovered although only their intensity (square modulus) is recorded. The information contained in several micrographs and an accurate theoretical descrip tion of the imaging properties of the electron microscope are combined into a pictorial form that is easily interpreted visually. Also, by careful consideration of the transfer function, the significant part of the transferred signal (transfer function large) may be separated from the noise (transfer function approaching or equal to zero), increasing the overall signal-to-noise ratio of the constructed image. The image reconstruction procedure does however impose several additional experiment difficulties as compared to conventional electron microscopy. It is necessary to record two to four additional micrographs (defocus series) of the same area of the specimen, increasing the total radiation dose to the specimen. All of these micrographs must be accurately aligned positionally once they reach the computer, and furthermore, they must all be of almost exactly the same magnification. Perhaps the most stringent additional requirement is the measurement of the electron optical properties of the microscope to a high accuracy (see ref. [19]). Although possible, accurate measurement of these parameters is almost superfluous in conventional electron microscopy. Needless to say, there is also the added inconvenience of maintaining a moderate computer facility in addition to an electron microscope facility. All of these additional experimental difficulties are in fact surmountable, although inconvenient. However, computer image reconstruction is the only way to obtain the complete complex scattered wave, which has been shown to have definite advantages in atomic structure research [11,12]. Also, because of the extreme expense and effort required to build better working high resolution microscopes and the decreasing cost of computers, computer image reconstruction may in fact

E.J. Kirkland et al. / Digital reconstruction o]'phase contrast hnages

prove to be cost-effective means of obtaining high resolution. For these reasons, the added experimental difficulties may well be worth the effort. An experimental defocus series (fig. 2) and a theoretically calculated defocus series [37] have been processed to demonstrate that the image reconstruction procedure developed in sections 2 and 3 is correct in theory and in practice. In sections 2 and 3 two nonlinear quadratic terms in the image model were shown to be small and neglected primarily to make the inaage reconstruction mathematics tractable. The theoretical simulation (numerical) in [37] has shown that neglecting these nonlinear terms leads to an overall quantitative error of less than 1-3%, which is visually undetectable. Therefore, the image reconstructed from the real micrographs (defocus series, fig. 2) in sections 4 and 5 may be interpreted with some degree of confidence. The signal-to-noise ratio, SfN, of the original micrographs was increased to 10 by signal averaging (multiple photographic exposures were made of many identical areas of the specinlen). This maintains a low radiation dose to prevent undue specimen damage. With this S I N (= I0) a resolution of 1.23 A can theoretically be expected from the defocus series shown in fig. 2. The reconstructed images (figs. 7 - 1 0 ) show a resolution of approximately 1.4 A. Many atomic features are clearly visible. However, there are several discrepancies between the atomic structure observed in the reconstructed image and the proposed atomic structure model (fig. 20. The most probable cause for this discrepancy is a slight misalignment of the crystal. The c-axis of the crystal may not be exactly parallel to the optical axis of the microscope causing a projection of dissimilar atomic positions into the final recorded image. Also, this specimen may be more radiation sensitive than supposed and the structure discrepancies may be radiation damage induced changes or the specimen may be somewhat thicker than supposed causing structure aliasing due to multiple scattering of the imaging electrons in the specimen. Most of the atoms in the structure model (fig. 2 0 may be accounted for in the reconstructed image (fig. 10) although the finely detailed atomic structure in both the real and imaginary parts of the reconstructed image is confusing. This fine structure is at the limit of resolution of the image reconstruction procedure for this microscope and thus can be expected to be difficult to interpret. Overall, the image

501

reconstruction procedure yields an increase in resolution (primarily in the real part of the image) and an increased "heavy/light" atom discrimination (in the imaginary part of the image). In retrospect, there are two potential problems that may have been treated too lightly. First, there is the problem of the tilt. No attention was paid to the direction or magnitude of the tilt when the experimental defocus series was processed (sections 4 and 5). It is not completely clear that a tilt of 26.5 ° is entirely benign. However, if the tilt has been ignored consistently half of the input micrographs (see fig. 2) can be expected to have their direction of tilt pointing up and the other half can be expected to have their direction of tilt pointing down (the symmetry of the molecule prevents any other combinations). That is, the top of half of the micrographs is a projection of the highest part (in z) of the specimen and the top of the other half of the micrographs is a projection of the lowest part of the specimen (and vice versa for the bottom of each micrograph). Because the image reconstruction (sections 2 and 3) is linear the whole defocus series may be considered collectively. The whole defocus series with a randomized tilt is in effect symmetric in z so that the analysis of thickness effects in section 4 applies. The tilt is identical to a small thickness (t = 17 A) effect which is negligible. The potential problem of the tilt may have been dealt with unknowingly. Secondly, the discrete nature of the picture (pixels) has been ignored In a discrete FFT (fast Fourier transform) the picture is actually treated as if it were infinitely periodic. The basic area (128 X 128 pixels in sections 4 - 6 ) is repeated indefinitely in two dimensions. The left-hand side and the right-hand side essentially wrap around to join each other (this same relation also exists between the top and the bottom of the image). When the image reconstruction is performed information is taken out of the left-hand edge of the image to reconstruct the right-hand edge of the image (with a characteristic distance equal to the average point spread function). Because these are not identical areas of the specimen spurious structure may be generated around the edges of the image. Therefore, the structure around the edges of the image should be ignored. Ideally, because this is a periodic specimen the leftand right-hand sides (also the top and bottom) of the image could be made identical. This would be compli-

502

E.J. Kirkland et al. / Digital reconstruction o f phase contrast images

cated because the unit cell is not square and the input images are inherently scanned on a square grid. However, most of the central area of the image containing more than one molecule is still a valid image reconstruction and the edges are not needed. Regardless of these experimental complications for this specimen, the overall performance of the MMSE image reconstruction procedure developed in section 2 is quite satisfactory and it is hoped that future applications of this method will also be fruitful.

Acknowledgements This work has been stipported by the National Science Foundation,USA (Grant No. ENG-7909093), the National Institutes of Health, USA (Grant No. GM-128738), and the Ministry of Education, Science and Culture, Japan (special equipment and Grants-inaid No. 843003 and No. 347012). Also one of the authors (E.J.K.) is especially thankful to Dr. P. Fejes for many interesting conversations concerning image formation.

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